From Lemma , we have that the induced subnetworks from the sPCAN algorithm is a network sample that satisfies \(G_{i,k} \sim \mathcal {G}(n_{i,k}, f_i)\)  for all \(k = 1, \ldots , K\) . The asymptotic result in (REF ) follows from the application of Theorem 1 of [1]} to the sample \(\lbrace G_{i, 1}, \ldots , G_{i, K}\rbrace \) . When the sample sizes of the networks in each bin are equivalent up to permutations, we furthermore know from [1]} that \(\Sigma _{\mathbf {n}_i,f_i,\mathbf {F}_p}\)  no longer depends on \(i\) . This proves the theorem.
